00:01
Hello, let pn be the set of all polynomials with real coefficients of degree less than equal to n.
00:06
In this question we have to prove that pn is a vector space.
00:09
Now, a general element, let f and g be general elements of pn, then f can be written as a0 plus a1x up to anx to the power n, and g can be written as b0 plus b1x up to bnx to the power n, and h be another general element which has the expansion c0, c1x up to cnx to the power n, which we can concisely represent in the summation notation.
00:36
Now, we want to show that the set of polynomials satisfies the axioms of a vector space.
00:42
The first axiom is that it should be closed under addition.
00:45
Now, when we add two polynomials, the addition in the set is the ordinary addition of polynomials.
00:52
So, when we add two polynomials, f plus g, we get the polynomial this thing, where the coefficients are the sum of the corresponding coefficients of f and g.
01:03
Now, this is again a polynomial of degree n, so it belongs to pn.
01:07
This is again a polynomial whose degree is less than equal to n.
01:11
We don't know that this degree is equal to n, because it may happen that an plus bn might be equal to 0, hence the degree may fall.
01:18
But the degree cannot be greater than n, so it has to be less than equal to n.
01:22
So, f plus g is a polynomial which will again be in pn.
01:25
Hence, pn is closed under addition.
01:27
Now, to prove associativity, let us take two, three general polynomials f, g, h, and let us consider the sum f plus g summed with h.
01:36
Now, f plus g is represented like this, and this plus h is represented like this.
01:40
So, the coefficients of f plus g plus h will be this thing, but addition of real numbers is associative.
01:47
So, this sum we can represent as this sum.
01:50
But this sum, this sum, and then this polynomial we can represent as the sum of this polynomial and this polynomial, which is equal to f plus gh.
02:01
So, we see that this sum is equal to this sum for three general elements of pn.
02:06
So, addition is associative.
02:08
Addition in pn is associative.
02:10
Now, for the polynomials f and g, if we look at the sum of ith coefficient, ai plus bi, it is also equal to bi plus ai.
02:24
But then, f plus g, which is equal to this polynomial, whose coefficients are ai plus bi, is also equal to the polynomial whose coefficients are bi plus ai.
02:36
But this polynomial is the sum of g plus f.
02:41
So, we see that f plus g is same as the polynomial g plus f.
02:44
So, the addition operation is commutative.
02:48
Now, given a polynomial f, consider the polynomial minus f, which we define to be sigma minus ai xi.
02:57
Now, f plus minus f is ai plus minus ai xi, which is equal to 0 xi, which is 0.
03:04
Note that in polynomials, the zero element for addition or the additive identity is the zero polynomial.
03:13
And we see that f plus minus f is the zero polynomial.
03:16
So, minus f is the unique additive inverse of f.
03:20
So, for every element, there exists an additive inverse in the set pn.
03:26
Now, let us focus on scalar multiplication...